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# rod mill 4 11

## ball mills

In all ore dressing and milling Operations, including flotation, cyanidation, gravity concentration, and amalgamation, the Working Principle is to crush and grind, often with rob mill & ball mills, the ore in order to liberate the minerals. In the chemical and process industries, grinding is an important step in preparing raw materials for subsequent treatment.In present day practice, ore is reduced to a size many times finer than can be obtained with crushers. Over a period of many years various fine grinding machines have been developed and used, but the ball mill has become standard due to its simplicity and low operating cost.

A ball millefficiently operated performs a wide variety of services. In small milling plants, where simplicity is most essential, it is not economical to use more than single stage crushing, because the Steel-Head Ball or Rod Mill will take up to 2 feed and grind it to the desired fineness. In larger plants where several stages of coarse and fine crushing are used, it is customary to crush from 1/2 to as fine as 8 mesh.

Many grinding circuits necessitate regrinding of concentrates or middling products to extremely fine sizes to liberate the closely associated minerals from each other. In these cases, the feed to the ball mill may be from 10 to 100 mesh or even finer.

Where the finished product does not have to be uniform, a ball mill may be operated in open circuit, but where the finished product must be uniform it is essential that the grinding mill be used in closed circuit with a screen, if a coarse product is desired, and with a classifier if a fine product is required. In most cases it is desirable to operate the grinding mill in closed circuit with a screen or classifier as higher efficiency and capacity are obtained. Often a mill using steel rods as the grinding medium is recommended, where the product must have the minimum amount of fines (rods give a more nearly uniform product).

Often a problem requires some study to determine the economic fineness to which a product can or should be ground. In this case the 911Equipment Company offers its complete testing service so that accurate grinding mill size may be determined.

Until recently many operators have believed that one particular type of grinding mill had greater efficiency and resulting capacity than some other type. However, it is now commonly agreed and accepted that the work done by any ballmill depends directly upon the power input; the maximum power input into any ball or rod mill depends upon weight of grinding charge, mill speed, and liner design.

The apparent difference in capacities between grinding mills (listed as being the same size) is due to the fact that there is no uniform method of designating the size of a mill, for example: a 5 x 5 Ball Mill has a working diameter of 5 inside the liners and has 20 per cent more capacity than all other ball mills designated as 5 x 5 where the shell is 5 inside diameter and the working diameter is only 48 with the liners in place.

Ball-Rod Mills, based on 4 liners and capacity varying as 2.6 power of mill diameter, on the 5 size give 20 per cent increased capacity; on the 4 size, 25 per cent; and on the 3 size, 28 per cent. This fact should be carefully kept in mind when determining the capacity of a Steel- Head Ball-Rod Mill, as this unit can carry a greater ball or rod charge and has potentially higher capacity in a given size when the full ball or rod charge is carried.

A mill shorter in length may be used if the grinding problem indicates a definite power input. This allows the alternative of greater capacity at a later date or a considerable saving in first cost with a shorter mill, if reserve capacity is not desired. The capacities of Ball-Rod Mills are considerably higher than many other types because the diameters are measured inside the liners.

The correct grinding mill depends so much upon the particular ore being treated and the product desired, that a mill must have maximum flexibility in length, type of grinding medium, type of discharge, and speed.With the Ball-Rod Mill it is possible to build this unit in exact accordance with your requirements, as illustrated.

To best serve your needs, the Trunnion can be furnished with small (standard), medium, or large diameter opening for each type of discharge. The sketch shows diagrammatic arrangements of the four different types of discharge for each size of trunnion opening, and peripheral discharge is described later.

Ball-Rod Mills of the grate discharge type are made by adding the improved type of grates to a standard Ball-Rod Mill. These grates are bolted to the discharge head in much the same manner as the standard headliners.

The grates are of alloy steel and are cast integral with the lifter bars which are essential to the efficient operation of this type of ball or rod mill. These lifter bars have a similar action to a pump:i. e., in lifting the product so as to discharge quickly through the mill trunnion.

These Discharge Grates also incorporate as an integral part, a liner between the lifters and steel head of the ball mill to prevent wear of the mill head. By combining these parts into a single casting, repairs and maintenance are greatly simplified. The center of the grate discharge end of this mill is open to permit adding of balls or for adding water to the mill through the discharge end.

Instead of being constructed of bars cast into a frame, Grates are cast entire and have cored holes which widen toward the outside of the mill similar to the taper in grizzly bars. The grate type discharge is illustrated.

The peripheral discharge type of Ball-Rod Mill is a modification of the grate type, and is recommended where a free gravity discharge is desired. It is particularly applicable when production of too many fine particles is detrimental and a quick pass through the mill is desired, and for dry grinding.

The drawings show the arrangement of the peripheral discharge. The discharge consists of openings in the shell into which bushings with holes of the desired size are inserted. On the outside of the mill, flanges are used to attach a stationary discharge hopper to prevent pulp splash or too much dust.

The mill may be operated either as a peripheral discharge or a combination or peripheral and trunnion discharge unit, depending on the desired operating conditions. If at any time the peripheral discharge is undesirable, plugs inserted into the bushings will convert the mill to a trunnion discharge type mill.

Unless otherwise specified, a hard iron liner is furnished. This liner is made of the best grade white iron and is most serviceable for the smaller size mills where large balls are not used. Hard iron liners have a much lower first cost.

Electric steel, although more expensive than hard iron, has advantage of minimum breakage and allows final wear to thinner section. Steel liners are recommended when the mills are for export or where the source of liner replacement is at a considerable distance.

Molychrome steel has longer wearing qualities and greater strength than hard iron. Breakage is not so apt to occur during shipment, and any size ball can be charged into a mill equipped with molychrome liners.

Manganese liners for Ball-Rod Mills are the world famous AMSCO Brand, and are the best obtainable. The first cost is the highest, but in most cases the cost per ton of ore ground is the lowest. These liners contain 12 to 14% manganese.

The feed and discharge trunnions are provided with cast iron or white iron throat liners. As these parts are not subjected to impact and must only withstand abrasion, alloys are not commonly used but can be supplied.

Gears for Ball-Rod Mills drives are furnished as standard on the discharge end of the mill where they are out of the way of the classifier return, scoop feeder, or original feed. Due to convertible type construction the mills can be furnished with gears on the feed end. Gear drives are available in two alternative combinations, which are:

All pinions are properly bored, key-seated, and pressed onto the steel countershaft, which is oversize and properly keyseated for the pinion and drive pulleys or sheaves. The countershaft operates on high grade, heavy duty, nickel babbitt bearings.

Any type of drive can be furnished for Ball-Rod Mills in accordance with your requirements. Belt drives are available with pulleys either plain or equipped with friction clutch. Various V- Rope combinations can also be supplied.

The most economical drive to use up to 50 H. P., is a high starting torque motor connected to the pinion shaft by means of a flat or V-Rope drive. For larger size motors the wound rotor (slip ring) is recommended due to its low current requirement in starting up the ball mill.

Should you be operating your own power plant or have D. C. current, please specify so that there will be no confusion as to motor characteristics. If switches are to be supplied, exact voltage to be used should be given.

Even though many ores require fine grinding for maximum recovery, most ores liberate a large percentage of the minerals during the first pass through the grinding unit. Thus, if the free minerals can be immediately removed from the ball mill classifier circuit, there is little chance for overgrinding.

This is actually what has happened wherever Mineral Jigs or Unit Flotation Cells have been installed in the ball mill classifier circuit. With the installation of one or both of these machines between the ball mill and classifier, as high as 70 per cent of the free gold and sulphide minerals can be immediately removed, thus reducing grinding costs and improving over-all recovery. The advantage of this method lies in the fact that heavy and usually valuable minerals, which otherwise would be ground finer because of their faster settling in the classifier and consequent return to the grinding mill, are removed from the circuit as soon as freed. This applies particularly to gold and lead ores.

Ball-Rod Mills have heavy rolled steel plate shells which are arc welded inside and outside to the steel heads or to rolled steel flanges, depending upon the type of mill. The double welding not only gives increased structural strength, but eliminates any possibility of leakage.

Where a single or double flanged shell is used, the faces are accurately machined and drilled to template to insure perfect fit and alignment with the holes in the head. These flanges are machined with male and female joints which take the shearing stresses off the bolts.

The Ball-Rod Mill Heads are oversize in section, heavily ribbed and are cast from electric furnace steel which has a strength of approximately four times that of cast iron. The head and trunnion bearings are designed to support a mill with length double its diameter. This extra strength, besides eliminating the possibility of head breakage or other structural failure (either while in transit or while in service), imparts to Ball-Rod Mills a flexibility heretofore lacking in grinding mills. Also, for instance, if you have a 5 x 5 mill, you can add another 5 shell length and thus get double the original capacity; or any length required up to a maximum of 12 total length.

On Type A mills the steel heads are double welded to the rolled steel shell. On type B and other flanged type mills the heads are machined with male and female joints to match the shell flanges, thus taking the shearing stresses from the heavy machine bolts which connect the shell flanges to the heads.

The manhole cover is protected from wear by heavy liners. An extended lip is provided for loosening the door with a crow-bar, and lifting handles are also provided. The manhole door is furnished with suitable gaskets to prevent leakage.

The mill trunnions are carried on heavy babbitt bearings which provide ample surface to insure low bearing pressure. If at any time the normal length is doubled to obtain increased capacity, these large trunnion bearings will easily support the additional load. Trunnion bearings are of the rigid type, as the perfect alignment of the trunnion surface on Ball-Rod Mills eliminates any need for the more expensive self-aligning type of bearing.

The cap on the upper half of the trunnion bearing is provided with a shroud which extends over the drip flange of the trunnion and effectively prevents the entrance of dirt or grit. The bearing has a large space for wool waste and lubricant and this is easily accessible through a large opening which is covered to prevent dirt from getting into the bearing.Ball and socket bearings can be furnished.

Scoop Feeders for Ball-Rod Mills are made in various radius sizes. Standard scoops are made of cast iron and for the 3 size a 13 or 19 feeder is supplied, for the 4 size a 30 or 36, for the 5 a 36 or 42, and for the 6 a 42 or 48 feeder. Welded steel scoop feeders can, however, be supplied in any radius.

The correct size of feeder depends upon the size of the classifier, and the smallest feeder should be used which will permit gravity flow for closed circuit grinding between classifier and the ball or rod mill. All feeders are built with a removable wearing lip which can be easily replaced and are designed to give minimum scoop wear.

A combination drum and scoop feeder can be supplied if necessary. This feeder is made of heavy steel plate and strongly welded. These drum-scoop feeders are available in the same sizes as the cast iron feeders but can be built in any radius. Scoop liners can be furnished.

The trunnions on Ball-Rod Mills are flanged and carefully machined so that scoops are held in place by large machine bolts and not cap screws or stud bolts. The feed trunnion flange is machined with a shoulder for insuring a proper fit for the feed scoop, and the weight of the scoop is carried on this shoulder so that all strain is removed from the bolts which hold the scoop.

High carbon steel rods are recommended, hot rolled, hot sawed or sheared, to a length of 2 less than actual length of mill taken inside the liners. The initial rod charge is generally a mixture ranging from 1.5 to 3 in diameter. During operation, rod make-up is generally the maximum size. The weights per lineal foot of rods of various diameters are approximately: 1.5 to 6 lbs.; 2-10.7 lbs.; 2.5-16.7 lbs.; and 3-24 lbs.

Forged from the best high carbon manganese steel, they are of the finest quality which can be produced and give long, satisfactory service. Data on ball charges for Ball-Rod Mills are listed in Table 5. Further information regarding grinding balls is included in Table 6.

Rod Mills has a very define and narrow discharge product size range. Feeding a Rod Mill finer rocks will greatly impact its tonnage while not significantly affect its discharge product sizes. The 3.5 diameter rod of a mill, can only grind so fine.

Crushers are well understood by most. Rod and Ball Mills not so much however as their size reduction actions are hidden in the tube (mill). As for Rod Mills, the image above best expresses what is going on inside. As rocks is feed into the mill, they are crushed (pinched) by the weight of its 3.5 x 16 rods at one end while the smaller particles migrate towards the discharge end and get slightly abraded (as in a Ball Mill) on the way there.

We haveSmall Ball Mills for sale coming in at very good prices. These ball mills are relatively small, bearing mounted on a steel frame. All ball mills are sold with motor, gears, steel liners and optional grinding media charge/load.

Ball Mills or Rod Mills in a complete range of sizes up to 10 diameter x20 long, offer features of operation and convertibility to meet your exactneeds. They may be used for pulverizing and either wet or dry grindingsystems. Mills are available in both light-duty and heavy-duty constructionto meet your specific requirements.

All Mills feature electric cast steel heads and heavy rolled steelplate shells. Self-aligning main trunnion bearings on large mills are sealedand internally flood-lubricated. Replaceable mill trunnions. Pinion shaftbearings are self-aligning, roller bearing type, enclosed in dust-tightcarrier. Adjustable, single-unit soleplate under trunnion and drive pinionsfor perfect, permanent gear alignment.

Ball Mills can be supplied with either ceramic or rubber linings for wet or dry grinding, for continuous or batch type operation, in sizes from 15 x 21 to 8 x 12. High density ceramic linings of uniform hardness male possible thinner linings and greater and more effective grinding volume. Mills are shipped with liners installed.

Complete laboratory testing service, mill and air classifier engineering and proven equipment make possible a single source for your complete dry-grinding mill installation. Units available with air swept design and centrifugal classifiers or with elevators and mechanical type air classifiers. All sizes and capacities of units. Laboratory-size air classifier also available.

A special purpose batch mill designed especially for grinding and mixing involving acids and corrosive materials. No corners mean easy cleaning and choice of rubber or ceramic linings make it corrosion resistant. Shape of mill and ball segregation gives preferential grinding action for grinding and mixing of pigments and catalysts. Made in 2, 3 and 4 diameter grinding drums.

Nowadays grinding mills are almost extensively used for comminution of materials ranging from 5 mm to 40 mm (3/161 5/8) down to varying product sizes. They have vast applications within different branches of industry such as for example the ore dressing, cement, lime, porcelain and chemical industries and can be designed for continuous as well as batch grinding.

Ball mills can be used for coarse grinding as described for the rod mill. They will, however, in that application produce more fines and tramp oversize and will in any case necessitate installation of effective classification.If finer grinding is wanted two or three stage grinding is advisable as for instant primary rod mill with 75100 mm (34) rods, secondary ball mill with 2540 mm(11) balls and possibly tertiary ball mill with 20 mm () balls or cylpebs.To obtain a close size distribution in the fine range the specific surface of the grinding media should be as high as possible. Thus as small balls as possible should be used in each stage.

The principal field of rod mill usage is the preparation of products in the 5 mm0.4 mm (4 mesh to 35 mesh) range. It may sometimes be recommended also for finer grinding. Within these limits a rod mill is usually superior to and more efficient than a ball mill. The basic principle for rod grinding is reduction by line contact between rods extending the full length of the mill, resulting in selective grinding carried out on the largest particle sizes. This results in a minimum production of extreme fines or slimes and more effective grinding work as compared with a ball mill. One stage rod mill grinding is therefore suitable for preparation of feed to gravimetric ore dressing methods, certain flotation processes with slime problems and magnetic cobbing. Rod mills are frequently used as primary mills to produce suitable feed to the second grinding stage. Rod mills have usually a length/diameter ratio of at least 1.4.

Tube mills are in principle to be considered as ball mills, the basic difference being that the length/diameter ratio is greater (35). They are commonly used for surface cleaning or scrubbing action and fine grinding in open circuit.

In some cases it is suitable to use screened fractions of the material as grinding media. Such mills are usually called pebble mills, but the working principle is the same as for ball mills. As the power input is approximately directly proportional to the volume weight of the grinding media, the power input for pebble mills is correspondingly smaller than for a ball mill.

A dry process requires usually dry grinding. If the feed is wet and sticky, it is often necessary to lower the moisture content below 1 %. Grinding in front of wet processes can be done wet or dry. In dry grinding the energy consumption is higher, but the wear of linings and charge is less than for wet grinding, especially when treating highly abrasive and corrosive material. When comparing the economy of wet and dry grinding, the different costs for the entire process must be considered.

An increase in the mill speed will give a directly proportional increase in mill power but there seems to be a square proportional increase in the wear. Rod mills generally operate within the range of 6075 % of critical speed in order to avoid excessive wear and tangled rods. Ball and pebble mills are usually operated at 7085 % of critical speed. For dry grinding the speed is usually somewhat lower.

The mill lining can be made of rubber or different types of steel (manganese or Ni-hard) with liner types according to the customers requirements. For special applications we can also supply porcelain, basalt and other linings.

The mill power is approximately directly proportional to the charge volume within the normal range. When calculating a mill 40 % charge volume is generally used. In pebble and ball mills quite often charge volumes close to 50 % are used. In a pebble mill the pebble consumption ranges from 315 % and the charge has to be controlled automatically to maintain uniform power consumption.

In all cases the net energy consumption per ton (kWh/ton) must be known either from previous experience or laboratory tests before mill size can be determined. The required mill net power P kW ( = ton/hX kWh/ton) is obtained from

Trunnions of S.G. iron or steel castings with machined flange and bearing seat incl. device for dismantling the bearings. For smaller mills the heads and trunnions are sometimes made in grey cast iron.

The mills can be used either for dry or wet, rod or ball grinding. By using a separate attachment the discharge end can be changed so that the mills can be used for peripheral instead of overflow discharge.

## grinding cylpebs

Our automatic production line for the grinding cylpebs is the unique. With stable quality, high production efficiency, high hardness, wear-resistant, the volumetric hardness of the grinding cylpebs is between 60-63HRC,the breakage is less than 0.5%. The organization of the grinding cylpebs is compact, the hardness is constant from the inner to the surface. Now has extensively used in the cement industry, the wear rate is about 30g-60g per Ton cement.

Grinding Cylpebs are made from low-alloy chilled cast iron. The molten metal leaves the furnace at approximately 1500 C and is transferred to a continuous casting machine where the selected size Cylpebs are created; by changing the moulds the full range of cylindrical media can be manufactured via one simple process. The Cylpebs are demoulded while still red hot and placed in a cooling section for several hours to relieve internal stress. Solidification takes place in seconds and is formed from the external surface inward to the centre of the media. It has been claimed that this manufacturing process contributes to the cost effectiveness of the media, by being more efficient and requiring less energy than the conventional forging method.

Because of their cylindrical geometry, Cylpebs have greater surface area and higher bulk density compared with balls of similar mass and size. Cylpebs of equal diameter and length have 14.5% greater surface area than balls of the same mass, and 9% higher bulk density than steel balls, or 12% higher than cast balls. As a result, for a given charge volume, about 25% more grinding media surface area is available for size reduction when charged with Cylpebs, but the mill would also draw more power.

## ivaco rolling mills

Ivaco Rolling Mills is a world-class producer of hot rolled wire rod and steel billets. Ourfacilities arelocated on the banks of the Ottawa River in LOrignal, Ontario between Montreal and Ottawa, two major Canadian cities. The Company was established in the early 1970s and was acquired by Heico Holdings Inc. in 2004.Ivaco Rolling Millsis dedicated tosupplyinghigh-quality products to boththe domestic and international markets and is proud to be a certified Canadian WBE business.

Ivaco Rolling Mills is driven by our mission to impress our customers with the quality, dependability and performance of our products and service. As needs are identified in the marketplace, programs are initiated to develop products, processes, equipment and services to meet these requirements.

Ivaco Rolling Mills is backed by the global strength and resources of our parent company, Heico. As a privately held American holding company devoted to manufacturing, construction and industrial services, Heico has the ability to make decisions that serve market needs and customers rather than shareholders. Ivaco Rolling Mills is also part of the The Heico Metal Processing Group. Members of The Heico Metal Processing Group include the following companies.

Ivaco Rolling Mills is proud to be certified as a WBE business. WBEs standard of certification is a meticulous process involving an in-depth review of the business that is seeking certification to confirm the business is at least 51% owned, operated and controlled by a woman or women. By including women-owned businesses among their suppliers, corporations and government agencies demonstrate their commitment to fostering diversity and the continued development of their supplier diversity programs.

## force analysis and curve design for laying pipe in loop laying head of wire rod mills | chinese journal of mechanical engineering | full text

Laying head is a high-precision engineering device in hot-rolled high speed wire rod production line. Previously research works are focused on the laying pipe wear-resisting. Laying pipe curve design method based on wire rod kinematics and dynamics analyses are not reported before. In order to design and manufacture the laying pipe, the motion and force process of the wire rod in the laying pipe should be studied. In this paper, a novel approach is proposed to investigate the force modeling for hot-rolled wire rod in laying pipe. An idea of limited element method is used to analysis and calculates the forces between laying pipe inner surface and wire rod. The design requirements of laying pipe curve for manufacturing are discussed. The kinematics and dynamics modeling for numerical calculation are built. A laying pipe curve equation is proposed by discussing design boundary conditions. Numerical results with different laying pipe curves design parameters are plotted and compared. The proposed approach performs good result which can be applied for laying pipe curve design and analysis for engineering application.

Manufacturing of wire rod is obtained by a high speed hot-rolled wire rod production line. As a schematic layout of the process flow shown in Figure1 [1], a high speed hot-rolled wire rod production line contains reheating furnace, a series of rough mills, intermediate mills, finishing mills, several shears, loopers and water boxes. The finishing milled wire rod passes through several water boxes with 3545m length. Then the hot wire rod is quenched and controlled reduction of temperature to 750900C. After the finishing rolling and water box, there is a pinch roll and laying head. With the help of a high speed rotated pipe, the laying head will change hot wire rods into convoluted form and laying the wire rod coils on the Stelmor conveyor. The coils on Stelmor conveyor will be conveyed to the next process. At the same time, Stelmor conveyor provides cooling air and the coils will be cooled and achieve the desired final properties on the Stelmor conveyor.

Laying head is one of a key and precision device in high speed production line of hot-rolled wires. Laying head device is located in water cooling section after the finishing rolling mill. The finishing mill is designed for a speed of 90120m/s for wire rod having a minimum diameter. Normally the mill produces series for steel wire rods diameter is between 4.5 and 16.0mm [2, 3]. Thus, the speed of linear wire rods pass through the laying head is very fast while the laying head is rotating. The hot wire rods are formed into loops out of the laying head and continuously laid on a Stelmor conveyor to send to the next processes [4, 5].

Figure2 shows the procedure of hot wire rods laying out and carrying on the Stelmor conveyor in the form of coils for air cooling. The finished hot-rolled wire rods pass through the rotated laying pipe to form coils as shown in Figure2(a). The coils lay down onto the Stelmor conveyor and be sent to the next process is shown in Figure2(b). In the Stelmor conveyor, cold air is blown from the bottom of the conveyor and the coils temperature is reduced, thus the color of the coils become dark before conveying to next process as shown in Figure2(c).

Laying out coils wire rod on the air cool convey: a coils wire out of the laying head machine; b hot coils wire on the air cool convey; c cold coils wire on the air cool convey (permission by Henan Jiyuan Iron & Steel Co., Ltd)

The device of laying head is composed by several parts as shown in Figure3 [1]. The input shaft (1) transmits rotation from motor. A guide pipe (3) is located at entrance of laying head device to guide the wire rod into the laying pipe (7). A pair of bevel gears (2 and 4) is assembled on the input shaft and output shaft (5). The output shaft (5) is connected with laying pipe holder (6). There are two bears located at point A and B and supporting the output shaft (5), point B to C is the cantilever part of the laying pipe holder (6). In which, laying pipe (see in Figure4) is the most important part for this laying head device. The diagram of a laying pipe can be seen in Figure4.

Normally the linear speed of wire rod feeding into laying head is very fast. The maximum feed speed for the minimum dimension wire rod (5.5 mm) is more than 100 m/s in the rolling scheme. The laying pipe is a 3D pipe with a complicated space curve. Some laying pipes are shown in Figure5(a) and (b).

The output shaft continually rotates, meanwhile the linear wire rod pass into the rotated laying pipe from the entry and throughout from the exit. In this process, the wire rod is formed to be looped shape as coils and fall down on the conveyer. The dimension of the formed coils is decided by the radius of the laying pipe exit [6,7,8].

Laying pipe is an important part in laying head device for hot rolled high speed wire rod production line. Most of the research works of laying pipe are presented by patents. Some mechanical structures for high speed laying head devices are proposed in Refs. [9,10,11]. The speed relationship between laying head and rolling mill is presented in Ref. [12] for control. Fiorucci tried different materials used for laying pipe in Refs. [13, 14], and some new design schemes to improve the wearing resistance property for laying pipes are reported in Refs. [15,16,17]. The manufacturing device and operating process for laying pipe is presented in Ref. [18]. A curve design and load analysis method is presented in Ref. [19], which is benefit for laying pipe applies theoretically. An optimal design of the Morgan Construction Companies laying pipe is investigated in Ref. [20] to deal with uneven wear. A pipe design approach is developed to minimize and evenly distribute wear on the inner surface of laying pipe. Some researchers have provided ideas to improve the laying hand performance in modeling, design and application [9, 21,22,23,24,25,26,27].

Most of the mentioned research work focused on the wear-resisting for laying pipe application. The wire rod kinematics and dynamics performance while passing through the laying pipe is not referred. Therefore, the design method based on wire rod kinematics and dynamics analysis for laying pipe curve cannot be carried out. Thus, the study on the topic of wire rod force analysis in laying pipe is a research gap before.

In this paper, a method for laying pipe design, as well as the kinematics and dynamics modeling for the wire rod passing in the laying pipe are proposed. The first section introduces the hot rolled high speed wire rod production line. The function and structure of the laying head machine is described. It is pointed out that the laying pipe is the key issue for the laying head. In the second section, design requirements for the laying pipe are discussed. The problems for pipe curve, laying head working stability and speed relationships are mentioned. In the third section, a method of kinematics and dynamics analysis for wire rod in the pipe is proposed to characterize the process of wire rod in the laying pipe. We also proposed a new method for laying pipe design in fourth section. The numerical computation results for the designed laying pipe are compared with existing laying pipes in the fifth section. The numerical results indicated the feasibility of the proposed design method and kinematics and dynamics modeling. It is also provided a method for laying pipe parameters choosing and discussing.

The wire rod is pinched and feed into rotated laying pipe by pinch roll as shown in Figure1. The wire rod in the laying pipe is effect by the pipe inner surface and its shape is changed from a straight line into coils. There are friction force, centrifugal force, inertial force and supporting force impacted on wire rod [28]. The entire forces act on the wire rod together and the wire is bended according the laying pipe curve and formed into coils to achieve the desired radius at the exit as Figure2. The requirements for the laying head pipe in a stable working condition can be explain from three aspects.

In the process of continuous rolling product line, it is necessary for laying head to match the speed relationship with pinch roll and finish mill. The linear speed of each machine should be maintained to match in a certain ratio in rolling process [29]. If the speed in one of the device is changed by the requirements of rolling production process flow, the speed of others devices should be changed relatively to maintain the matched relationship. In order to lay out coils stably and continually, the laying head will always follows the speed with pinch roll and finish mill. If the matched speed relationship cannot be kept, there would be an accident on the high speed wire rod product line. The hot-rolled wire rod will be pulled into two pieces or push toghter and be crowded out of the passing line as shown in Figure6.

A producing accident: wire rod pushed toghter and crowded out of the passing line, caused by speed relationship mismatched: a wire rod out of the passing line; b the pushed toghter and crowded wire rod

Another speed matching should also be maintained for the wire rod feed in and out the laying pipe. The feeding speed of wire rod into laying pipe is defined as vf, the radius at the exit of the laying pipe is defined as R0, 0 is the rotation speed of laying head output shaft. According the law of equal metal mass flow per second [29], there should be a balance relationship between the metal in and out the laying pipe which can be expressed as

The curve of laying pipe should be designed smoothly for the wire rod passing through easily. The requirement of the coils diameter is ensured by the exit position of the laying pipe. The speed of coils leave the laying pipe should be very small to ensure the coils can fall on the conveyer freely as shown in Figure7.

The laying pipe should be used for long term and cannot be worn easily. The worn is caused by the friction force on the inner surface of laying pipe. The curve should be designed smoothly to prevent the wire rod be worn sharply in some position. An ideal laying pipe curve can make the wire rod pass through comfortably. In this situation, the friction force will be decreased and will not be concentrated in small section of the laying pipe. As the result, the laying pipe will be used for a long term and the laying head working stability is improved.

What we desire is an arm exoskeleton which is capable of following motions of the human upper-limb accurately and supplying the human upper-limb with proper force feedback if needed. In order to achieve an ideal controlling performance, we have to examine the structure of the human upper-limb.

The origin of the cylindrical coordinates 0RZ is set at the entrance of laying pipe. In which, and R are described in a polar coordinates plane. Thus, the pipe curve $$\varvec{\delta}(R,\theta ,Z)$$ in this cylindrical coordinate can be formulated as

where is the rotated angle in the plane of R; R is the displacement of radial direction in the plane of R; Z is the displacement of axial direction in the cylindrical coordinate; T0 is the boundary condition for the variable parameter t.

The wire rod will pass through the laying pipe according the curve of Eq. (2), the velocity of the wire rod motion in the laying pipe can be deduced from the laying pipe curve. The wire rod in the pipe is selected as the object for study, the laying pipe is considered as a relative coordinate system. Thus, the velocity of a point of the wire rod in the pipe can be expressed as vector of $$\vec{\varvec{v}}_{\text{T}}$$,

where $$\vec{\varvec{e}}_{r}$$, $$\vec{\varvec{e}}_{\theta }$$, $$\vec{\varvec{e}}_{\text{z}}$$ represent unit direction vectors in radial direction r, tangential direction and axial direction z; vr, v, vz are the velocity in each direction respectively.

According the law of equal metal mass flow per second [29], the relative velocity between the wire rod and laying pipe is always equal to the feed speed vf. The unit vector of the velocity vector $$\vec{\varvec{v}}_{\text{T}}$$ in Eq. (3) can be expressed as

where $$\left| {\vec{\varvec{v}}_{\text{T}} } \right| = (v_{r}^{2} + v_{\theta }^{2} + v_{z}^{2} )^{1/2}$$. Thus, the relative velocity between the wire rod and laying pipe can be expressed in vector form as

where (vr)g,(v)g,(vz)g are the components of $$\vec{\varvec{v}}_{\text{g}}$$ in $$\vec{\varvec{e}}_{r} ,\;\vec{\varvec{e}}_{\theta } ,\;\vec{\varvec{e}}_{z}$$ directions respectively. In order to obtain the relative acceleration, take the derivative of Eq. (6) with parameter t at first. The relative acceleration of wire rod motion $$\vec{\varvec{a}}_{\text{g}}$$ can be expressed in form of

It should be noticed that the motion of the wire rod in tangential direction $$\vec{\varvec{e}}_{\theta }$$ is circular motion. Thus, there should be a centripetal acceleration in the radial direction of $$\vec{\varvec{e}}_{r}$$, which can be formulated

When the laying pipe curve Eq. (2) is given, the relative velocity between wire rod and laying pipe $$\vec{\varvec{v}}_{\text{g}}$$ can be calculated by Eq. (6) and the relative acceleration $$\vec{\varvec{a}}_{\text{g}}$$ can be obtained by substituting Eq. (11) into Eq. (10).

The relative motion for wire rod in laying pipe is analyzed from Eq. (2) to Eq. (11) in the moving coordinates system 0RZ. In order to obtain the wire rod motion in the fixed coordinates system, the rotated motion of the moving coordinates 0RZ should be analyzed. If the laying pipe rotates at a constant speed 0, it means the coordinates 0RZ is rotating with the speed 0. According motion synthesis principle, the convected motion velocity $$\vec{\varvec{v}}_{\text{e}}$$ and acceleration $$\vec{\varvec{a}}_{\text{e}}$$ of the laying pipe can be expressed as

Because the convected motion of the laying pipe is rotation movement, there is a Coriolis acceleration $$\vec{\varvec{a}}_{\text{c}}$$ in the system, which can be formulated according the definition of Coriolis acceleration as

Equation (15) express the absolute acceleration components in the three directions of the cylindrical coordinate in Figure8. The parameters of (ar)a, (a)a, (az)a are the target of the kinematics calculation.

A force state of the wire rod is modeling and analyzed in this part. In order to analysis and calculate the force affection of the wire rod, the entire wire rod in the laying pipe is divided into limited sectional parts. Each part unit can be seen as an independent object with several kinds of forces acted on it. Thus, a limited elements method is proposed and applied to build the dynamics modeling of sectional wire rod unit.

The wire rod passing through the laying pipe can be seen composed by limited sectional rod units, each rod unit is assumed as a straight rod. Each of the straight rod has its unique position in the laying pipe and the divided sectional rod units are connected by its axial force in the connected surface. All the sectional units are pulled or pushed by the axial force and pass through the laying pipe. A force modeling of two units of wire rods is shown in Figure9. If the divided sectional units are small enough, the results of this model is reasonable and available for the practical force situation.

In Figure9, $$\vec{\varvec{f}}_{mi}$$ is the friction force on the sectional rod unit N, acted by inner surface of laying pipe; $$\vec{\varvec{P}}_{ni}$$ is the supporting force on the section rod unit N, acted by inner surface of laying pipe; $$\vec{\varvec{N}}_{i}$$ is the axial push force on the section rod unit N, acted by the section rod unit (N1); $$\vec{\varvec{N}}_{i + 1}$$ is the pull force on the section rod unit N, acted by the section rod unit (N+1); $$\vec{\varvec{f}}_{i}$$ is the inertia force on the section rod unit N.

In Eqs. (16) and (17), mi is the mass of the section rod unit N (AB); fri, fi and fzi are the component forces in each direction; (ar)ai, (a)ai and (az)ai are the acceleration in each directions, also mentioned in Eq. (15).

where $$\sum \vec{\varvec{F}}$$ means the resultant force on the rod unit N. Eq. (18) can be divided into two directions, which are tangential force and normal force respectively. The equilibrium equations can be expressed as

where $$\text{(}\vec{\varvec{f}}_{i} )_{t}$$ is the tangential component of the inertia force on the section rod unit N; $$\text{(}\vec{\varvec{f}}_{i} )_{n}$$ is the normal component of the inertia force on the section rod unit N; $$(\vec{\varvec{N}}_{i + 1} )_{t}$$ is the tangential component of the pull force on the section rod unit N; $$(\vec{\varvec{N}}_{i + 1} )_{n}$$ is the normal component of the pull force on the section rod unit N.

In Eq. (19), the direction of the supporting force $$\vec{\varvec{P}}_{ni}$$ is the same as the normal of vector of section rod unit N (vector $$\overrightarrow {{\varvec{AB}}}$$), the direction of the friction force $$\vec{\varvec{f}}_{mi}$$ is the same as the tangential of vector $$\overrightarrow {{\varvec{AB}}}$$.

The tangential and normal components of inertia force $$\vec{\varvec{f}}_{i}$$ can be solved by using the directional cosine of the straight sectional unit N (vector $$\overrightarrow {{\varvec{AB}}}$$). The unit vector in direction of vector $$\overrightarrow {{\varvec{AB}}}$$ can be expressed as

where cos, cos and cos are the directional cosine of $$\vec{\varvec{e}}_{r} ,\;\vec{\varvec{e}}_{\theta } ,\;\vec{\varvec{e}}_{z}$$ in the cylindrical coordinates 0RZ. If the position of the sectional unit AB is known as vector $$\overrightarrow {{\varvec{AB}}}$$, the direction cosine cos, cos and cos can be calculated.

It can be concluded that the inertia force $$\vec{\varvec{f}}_{{i}}$$ can be calculated by providing the equation of laying pipe curve and the rotation speed of the laying head. Thus, the contact force $$\vec{\varvec{P}}_{ni}$$, friction force $$\vec{\varvec{f}}_{{mi}}$$ and push or pull force $$\vec{\varvec{N}}_{{i}}$$ for the sectional unite in the laying pipe can be calculated by Eqs. (27), (28) and (29), if the vector force $$\vec{\varvec{N}}_{{i+1}}$$ is known as an initial parameter.

For the proposed dynamics modeling of the wire rod in the laying pipe, boundary conditions are necessary to solve the force equilibrium Eq. (19). Boundary conditions for this modeling can be found at the entry and exit of the laying pipe as shown in Figure10.

There is a speed relationship between the wire rod feeding speed vf and the rotated speed of laying pipe 0 at the entry of the laying pipe, which has been formulated as vf=0R0 in Eq. (1) of Section2. The wire rod at the entry of the laying pipe should keep the relationship in Eq. (1) and it can be a boundary conditions at the entrance of the laying pipe.

As shown in Figure10, the wire rod sectional unit at the laying pipe exit will not push or pull its adjacent units along their centerline of the wire rod. There is only gravity force impacted on the wire coils at the pipe exit to make the coils fall on the conveyer freely and slowly. Thus, the pull force on the final section unit N of the wire rod is zero, which can be expressed as

Lets select the sectional unit at the exit of laying pipe to study, at this unique moment $$\vec{\varvec{N}}_{N + 1}$$ is a boundary condition for Eq. (19) as expressed in Eq. (30). Thus, the force balance Eq. (19) of the final unit can be solved. For rod unit N, the result of push force $$\vec{\varvec{N}}_{N}$$ in Eq. (19) can be calculated and the result $$\vec{\varvec{N}}_{N}$$, seen as the pull force for rod unit N1, can be used as a given parameter in the rod unit N1 force balance equation. Thus, all the forces on each sectional unit can be calculated by providing computed result $$\vec{\varvec{N}}_{i}$$ as a given parameter for i1 unit. Thus, the sectional units force equilibrium equations (19) will be solved one by one from the final sectional unit to the first sectional unit.

A numerical calculation procedure for the proposed modeling can be carried out based on the above-mentioned kinematics and dynamics analysis to compute the velocity and forces of the wire rod in the laying pipe.

In order to solve the velocity and forces of the wire rod, some parameters should be given as the initial conditions for the numerical procedure calculation. The curve equation $$\varvec{\delta}(R,\theta ,Z)$$, the feed speed of the linear wire rod vf and the rotation speed of the laying pipe 0 should also be given as the parameters for the kinematics and dynamics modeling.

It is assumed that the wire rod in the laying pipe always moves along the center of the laying pipe. Thus, the laying pipe curve equation can be used to describe the wire rod path in the laying pipe. The numerical procedure for the wire rod kinematics and dynamics calculation flowchart is shown in Figure11. In this flowchart, the laying pipe curve equation $$\varvec{\delta}(R,\theta ,Z)$$ , the feed speed of the linear wire rod vf before laying head and the rotation speed of the laying pipe 0 are given as initial conditions. Another boundary condition is the pull force $$\vec{\varvec{N}}_{i+1}$$ of the final sectional unit (unit N) which is equal to zero at the exit (mentioned in Eq. (30)). The flowchart of this numerical procedure can output the relative velocity $$\vec{\varvec{v}}_{\text{g}}$$ and acceleration $$\vec{\varvec{a}}_{\text{g}}$$ of the wire rod in laying pipe as well as the connect force $$\overrightarrow {\varvec{P}}_{ni}$$, the friction force $$\vec{\varvec{f}}_{mi}$$ and the axial force for push or pull in the wire rod centerline.

The curve of the laying pipe $$\varvec{\delta}(R,\theta ,Z)$$ are described in cylindrical coordinate 0RZ shown in Figure8. The pipe curve in this cylindrical coordinate can be formulated as Eq. (2), and Eq. (2) can also been seen as the description equation for wire rod motion path in the laying pipe:

where the design parameters for the laying pipe curve are as follows: T0 is the time for a wire rod point passing through the laying pipe; R0 is the radius dimension of laying pipe at the exit; 0 is the rotation angle of laying pipe curve at the exit; Z0 is the dimension of coordinate Z for laying pipe exit.

Actually, according our experience, when we focused on the exit of the laying pipe, dZ/dt is not exactly equal to 0. There should be a small angle at the laying pipe exit according the plane R-, which is defined as * o and can be shown in Figure4. The small angle * o is used to ensure the wire coils out of the laying pipe easily and could fall down on the conveyor with very small velocity. Usually, the angle $$\alpha_{0}^{*} \in [1.6^\circ ,\,2.5^\circ ]$$. Thus, the boundary conditions for the laying pipe curve design can be developed as

The boundary conditions in Eqs. (33) and (34) should satisfy the proposed laying pipe curve equation. These known conditions will help us to choose the form and parameters for laying pipe curve equation.

Eq. (39) qualified the characters of the laying pipe curve both in the entry and exit. The boundary conditions for the laying pipe curve in Eq. (33) are also ensured by Eq. (39). According to Eqs. (38) and (39), if t=T0, the equation for function Z(t) can be expressed as

The design parameters in Eq. (39) for laying pipe curve equation are decided by rolling production process, such as formed coils diameter and laying head structure. Most of these parameters have been fixed by milling process in a specific high speed wire rod production line. One of solutions for laying pipe design parameters in a high speed wire rod production line are given and list in Table1.

In order to make comparison with other laying pipes, two laying pipes using in different types laying head device are selected. The pipe curves are obtained by reverse engineering with laser 3D scanning. The laser scanning device is shown in Figure13(a), the obtained laying pipe modeling are shown in Figure13(b) and (c). The laying pipe curves in Figure13(b) and (c) is named 2 and 3.

R2(t) and R3(t) in Eqs. (42) and (43) are obtained by a series of coordinates data with t and R, which are listed in Table2. The coordinates data are used to fit functions for R2(t) and R3(t). The of functions R2(t) and R3(t) are obtained by 10 pieces of piecewise cubic spline curve composed functions.

A modeling results with the three laying pipe curves coordinates of 1, 2 and 3 are shown and compared in Figure14. Comparing with the other two laying pipe curves, the designed laying pipe curve 1 is much smooth between the entry and middle part of the curve.

Numerical computation are carried out for the three laying pipe curves 1, 2 and 3 by using the kinematics and dynamics model proposed in Section3. The calculation procedure is following the flowchart in Figure11. The wire rod diameter used for calculation is $$\phi$$6.5 mm, the feed speed of wire rod is 92 m/s. Some numerical results are shown from Figure15, 16, 17 Figure18.

Numerical results for wire rod relative velocity in different laying pipes: a relative velocity in laying pipe curve 1; b relative velocity in laying pipe curve 2; c relative velocity in laying pipe curve 3

Figure15 shows the numerical results for wire rod in laying pipes. It can be seen that the wire rod relative velocities in three laying pipes have the same tendency. The relative velocities in axial direction vz are all equal to the feed speed vf at the entry, and decrease meanwhile the wire rod is passing in the laying pipe. Finally it is nearly zero at the exit. The relative velocities in tangential direction are v are all zero at the entry, and change to nearly feed speed vf at the exit of the laying pipe. The relative velocities in radial direction are vr are all zero at the entry and change to the maximum in the middle part of the laying pipe, then it is decreased to zero at the exit of laying pipe.

Comparing with 2 and 3, the maximum value of v for designed laying pipe is appeared nearly the exit of the pipe. It is caused by the smooth curve between the pipe entry and middle part, and the wear of the laying pipe inner surface will be reduced in this position. The variable characters of the wire rod relative velocity in Figure15 ensure the laying pipes has only axial velocity at entry and has only tangential velocity at exit. The resultant velocity of the three velocities components is equal to the feed speed vf at any time in Figure15.

The dynamic numerical results for wire rod motion in laying pipes are shown from Figure16 to Figure18. Figure16 indicates that the axial force for wire rod in the three laying pipes. The tendency of axial forces is 3343MPa at the pipe entry, and increases to the maximum after 1.01.2 m then the axial forces decrease to zero at the exit of laying pipe. The maximum value axial force in laying pipe 1 is 48MPa; the maximum value axial forces in laying pipe 2 and 3 are 36 MPa.

Figure17 and Figure18 are the support forces and friction forces for the wire rod passing in the three laying pipes. According the Coulombs law of friction, the relationship between friction force and support force are proportional. The maximum friction force in unit length is 0.78 N/mm for the designed laying pipe 1. It appears at 1.35 m after laying pipe entry. The position with maximum friction force will be the worn out position when the laying pipe is used. Comparing with laying pipes 2 and 3, the worn out position in 1 is farther away from pipe entry than others. Because the designed laying pipes curve 1 is much smooth in the middle part than 2 and 3 in Figure14. Thus, the maximum friction position is changed.

In this section, some different values of parameters are changed for the designed laying pipe curve. Numerical results are carried out to discuss the dynamic performance with different design parameters. The initial design parameters are list in Table1, and the wire rod dimension is 6.5 mm.

Figure19 shows the wire rod axial force and friction force performance with different feed speed vf. According the law of equal metal mass flow per second, the relationship between the feed speed vf and rotation speed of laying head output shaft 0 formulated in Eq. (1) should be kept. Thus, the numerical results for vf equals 70 m/s, 90 m/s and 110m/s should match the 0 with 1273 r/min, 1637 r/min and 2000 r/min, respectively.

It can be seen that the maximum value of axial force in Figure19(a) are 30 MPa, 43 MPa and 72 MPa, respectively for different feed speeds. The axial force is increased obviously when the feed speed is 110 m/s. In Figure19(b), the maximum friction force in 110 m/s is about twice as much as 90 m/s. Comparing with other feed speed, when the feed speed is 110/m, the laying pipe will be worn out first. Actually, the higher feed speed will lead to fast wearing out, it has been verified in other wire rod rolling plant.

It should be noticed that, the tensile strength limitation for most hot-rolled rod materials are less than 70 MPa in the temperature range of 750900C. Thus, the wire rod will be pulled broken into pieces if the laying pipe curve working with vf =110 m/s. Thus, there should be a new designed laying pipe curve for the higher feed speed and rotation speed.

According to Figure20(a) and (b) the axial force and friction force will be less if the axial dimension Z0 is increasing. Thus, increasing the axial dimension Z0 will be benefit for the wire rod force impaction in the laying pipe. On the other hand, enlarge laying pipe axial dimension Z0 will lead the pipe holder cantilevers out too far as shown in Figure3. Thus, the dynamic balance will be difficult for the output shaft rotating.

Figure21 shows the wire rod axial force and friction force performance with different laying pipe curve rotated angle 0. It can be seen from Figure21(a) that the axial forces will increase if the curve rotated angle 0 is increased. The friction force will decrease if the curve rotated angle 0 is increased in Figure21(b). Thus, it can be concluded that increasing the curve rotated angle 0 will be benefit for reducing the friction force of the laying pipe, meanwhile the axial force should not more than the tensile strength limitation for the wire rod. The rotated angle 0[350, 400] is a good choice for pipe design.

Figure22 shows the wire rod axial force and friction force performance with wire rod diameters of 6.5 mm and 10mm respectively. When the production of wire rod diameter is 10 mm, the feed speed vf changes to 68 m/s according to the production process flow. The rotation speed 0 for laying pipe is also changed to keep the relationship in Eq. (1). It is indicated from Figure22(a) and (b) that the axial force and friction force for wire rod 6.5 mm are enhanced than wire rod 10 mm. Thus, it is confirmed that with the higher feed speed and rotation speed, the laying pipe for wire rod 6.5 mm is easier to be worn off than 10 mm.

It is indicated from the numerical results from Figure19 to Figure22 that the requirement for higher speed laying pipe curve design should increase the laying pipe axial dimension Z0 and choose a suitable pipe curve rotated angle 0 between 350 and 400.

The paper focused on the laying pipe of laying head device in the hot-rolled high speed wire rod production line. The kinematics and dynamics modeling are first proposed and investigated in this paper with the aim to obtain the wire rod performance while passing through the laying pipe. The statics study and calculation for the wire rod in the laying pipe are carried out by using sectional divided method. A laying pipe curve equation is formulated by considering its design boundary conditions. Thus, numerical results of wire rod velocity and forces can be calculated by the obtained laying pipe equations. With the proposed modeling, kinematics and dynamics results comparison between the designed laying pipe curve and two existing laying pipes are carried out. Some variable parameters of laying pipe curve are calculated with different value, and the results are discussed to find suitable design pipe curve parameters for different working conditions. The results of calculation and discussion indicate the proposed method of laying pipe curve design and the research for kinematics and dynamics are feasible for laying pipe application.

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SY conceived and studied the laying pipe design method and the force computations. MC contributed the force analysis of the laying pipes. Bin Ma performed the experiments for laying pipe curves. SY and MC wrote the paper. MC, GC and BM reviewed and edited the manuscript. All authors read and approved the final manuscript.

Shuangji Yao, born in 1981, he is currently an associate professor at Yanshan University, China. He received his PhD degree from Beihang University, China, in 2010. He has joined the study program at LARM during 20072008 in Italy. His research interests include the theory of mechanisms, robots, and mechanical research in rolling machine. He is a member of IEEE and ASME.

Marco Ceccarelli, born in 1958, received his Ph.D. degree in Applied Mechanics in 1988. He is a Full-Time Professor of Mechanics of Machinery and Director of LARM, Laboratory of Robotics and Mechatronics at University of Cassino and South Latium, Italy. He is a Member of Robotics Commission of IFToMM (The International Federation for the Promotion of Machine and Mechanism Science). He is the IFToMM President. He has written the books Fundamentals of Mechanics of Robotic Manipulation in 2004 and Mecanismos in 2008 and 2014. His research interests are in mechanics of mechanisms and robots. He is the author/co-author of more than 700 papers, presented at conferences or published in journals, and he has edited 23 books as for conference proceedings and specific topics.

Giuseppe Carbone, born in 1972, is currently an associate professor at University of Cassino and South Latium, Italy. He received his PhD degree from University of Cassino and South Latium, Italy. He has carried out several periods of research abroad, such as in Germany, Japan, Spain, and China. His research interests include stiffness of multibody robotic systems, robotic hands and grippers, mechatronic designs, design of experimental test-beds. He has published more than 250 peer reviewed papers on the above-mentioned topics.

We would especially like to thank Dingxuan Zhao, Wantang Fu and Qingtian Zhou for the fruitful discussions, leading to very interesting formulations of problems and deeper insight. Furthermore but not least, we would like to thank Zhifeng Pang, Yongqian Zhao and Renquan Wang for their great help and kindness.